Solution with a solve path (Solve it yourself on [Penpa+](https://tinyurl.com/274qe4tw))

Let's construct some rules to help us solve. 

- Corner rule: in the following configuration, where `*` denotes no connection and ` ` denotes an unknown connection
```
    B
     
X * A   C
    *
    Y 
```
then the cells ("in the corner") must be connected, so you can draw the two line segments B-A and A-C, giving
```
    B
    |
X * A - C
    *
    Y 
```

Applying the corner rule and easy logic to the starting position gets us

(1)
>! [![1][1]][1]

The bottom left can only be resolved like this. Use the corner rule as well. 

(2)
>! [![2][2]][2]

Invalid configuration rule: if there is a cell only reachable from one neighbour, then we have an invalid configuration. 

Short loop avoidance rule: in the following configuration
```
    B
     
X   A   C
    *
    Y 
```
where X and C belong to the same short loop, we cannot have X-A and A-C. Therefore B-A is connected (to avoid triggering the invalid configuration rule). So
```
    B
    |
X   A   C
    *
    Y 
```

Using this, we can make some deductions. 

(3) (The red segment leads to a quick contradiction and must be removed)
>! [![3][3]][3]

(4)
>! [![4][4]][4]

Connecting the red line leads to a contradiction in the top right after around 10 moves. So we have

(5)
>! [![5][5]][5]

Connecting the red line leads to a contradiction in the middle right after around 11 moves. So we have

(6)
>! [![6][6]][6]

Connecting some obvious things (starting from the bottom) leads to

(7)
>! [![7][7]][7]

Avoiding a big loop on the left leads to the solution

(8)
>! [![8][8]][8]


  [1]: https://i.sstatic.net/iVkPJ29j.png
  [2]: https://i.sstatic.net/Egiv1cZP.png
  [3]: https://i.sstatic.net/6jlkEXBM.png
  [4]: https://i.sstatic.net/WiCHoRGw.png
  [5]: https://i.sstatic.net/IrLvAfWk.png
  [6]: https://i.sstatic.net/oTZUyl3A.png
  [7]: https://i.sstatic.net/gYR8glaI.png
  [8]: https://i.sstatic.net/CUU0Z8wr.png